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Properties

Label 16.12.3393001956...1409.1
Degree $16$
Signature $[12, 2]$
Discriminant $17^{14}\cdot 67^{4}$
Root discriminant $34.13$
Ramified primes $17, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1194

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-307, -1082, 6343, 12503, -9109, -16107, 4109, 5676, -601, -3, -211, -333, 97, 65, -16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 16*x^14 + 65*x^13 + 97*x^12 - 333*x^11 - 211*x^10 - 3*x^9 - 601*x^8 + 5676*x^7 + 4109*x^6 - 16107*x^5 - 9109*x^4 + 12503*x^3 + 6343*x^2 - 1082*x - 307)
 
gp: K = bnfinit(x^16 - 4*x^15 - 16*x^14 + 65*x^13 + 97*x^12 - 333*x^11 - 211*x^10 - 3*x^9 - 601*x^8 + 5676*x^7 + 4109*x^6 - 16107*x^5 - 9109*x^4 + 12503*x^3 + 6343*x^2 - 1082*x - 307, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 16 x^{14} + 65 x^{13} + 97 x^{12} - 333 x^{11} - 211 x^{10} - 3 x^{9} - 601 x^{8} + 5676 x^{7} + 4109 x^{6} - 16107 x^{5} - 9109 x^{4} + 12503 x^{3} + 6343 x^{2} - 1082 x - 307 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3393001956715501807791409=17^{14}\cdot 67^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.13$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $17, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{137} a^{14} - \frac{29}{137} a^{13} + \frac{20}{137} a^{12} - \frac{45}{137} a^{11} + \frac{46}{137} a^{10} + \frac{67}{137} a^{9} - \frac{15}{137} a^{8} - \frac{33}{137} a^{7} + \frac{10}{137} a^{6} - \frac{59}{137} a^{5} + \frac{64}{137} a^{4} + \frac{65}{137} a^{3} - \frac{30}{137} a^{2} - \frac{22}{137} a + \frac{26}{137}$, $\frac{1}{22107330172198577525701} a^{15} - \frac{4760593233771745845}{22107330172198577525701} a^{14} + \frac{10182947996648536112767}{22107330172198577525701} a^{13} - \frac{6132604297597166077511}{22107330172198577525701} a^{12} - \frac{8392459118004194646163}{22107330172198577525701} a^{11} + \frac{7142647086945175234398}{22107330172198577525701} a^{10} + \frac{4998274055035421808341}{22107330172198577525701} a^{9} - \frac{5598102165353341437934}{22107330172198577525701} a^{8} + \frac{5109806976296007861093}{22107330172198577525701} a^{7} - \frac{9926904905639002283050}{22107330172198577525701} a^{6} - \frac{7816278529214726678501}{22107330172198577525701} a^{5} + \frac{4479444602322790041628}{22107330172198577525701} a^{4} + \frac{10414525533999129540409}{22107330172198577525701} a^{3} + \frac{10740060647942950542456}{22107330172198577525701} a^{2} + \frac{2619344542001272424742}{22107330172198577525701} a - \frac{6946512992116101065108}{22107330172198577525701}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4112173.95016 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1194:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1194
Character table for t16n1194 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
67Data not computed